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A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x. 16
1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 22, 12, 1, 1, 27, 92, 92, 27, 1, 1, 58, 359, 604, 359, 58, 1, 1, 121, 1311, 3607, 3607, 1311, 121, 1, 1, 248, 4540, 19912, 31238, 19912, 4540, 248, 1, 1, 503, 15110, 102842, 244424, 244424, 102842, 15110, 503, 1, 1, 1014 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

FORMULA

E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - Vladeta Jovovic, Apr 06 2001

T(n,k) = A123125(n,k) + A123125(n,k+1), with A123125(n,n+1) = 0. - Franck Maminirina Ramaharo, Oct 21 2018

EXAMPLE

Triangle begins:

   1;

   1,   1;

   1,   2,    1;

   1,   5,    5,    1;

   1,  12,   22,   12,    1;

   1,  27,   92,   92,   27,    1;

   1,  58,  359,  604,  359,   58,   1;

   1, 121, 1311, 3607, 3607, 1311, 121, 1;

   ...

MATHEMATICA

t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = (n-k) t[n-1, k-1] + (k+1) t[n-1, k];

A[n_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k];

T[n_, k_] := A[n, k] + A[n, k+1];

Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, May 26 2019, after Franck Maminirina Ramaharo *)

CROSSREFS

Cf. A008292.

Columns include A000325. Row sums are in A098558.

T(2n,n) gives A177042.

Sequence in context: A196636 A196641 A197090 * A264862 A176420 A099927

Adjacent sequences:  A008515 A008516 A008517 * A008519 A008520 A008521

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Vladeta Jovovic, Apr 06 2001

STATUS

approved

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Last modified May 21 05:37 EDT 2022. Contains 353889 sequences. (Running on oeis4.)